Decision making is a most common human activity. Individuals and organisations make all kinds of decisions in a variety of ways on a regular basis. Most decision problems are associated with a number of criteria, which may be measured in different ways, be in conflict with one another, and comprise both a quantitative and qualitative nature. In many situations, decision makers may have to make decisions on the basis of incomplete or partial information. For instance, buying a car may be an individual or a family decision and a customer will not buy a car without taking into account several criteria such as price, safety measures, size of engine, and general quality. Similarly, a company often will not do business with a supplier without assessing many criteria such as financial stability, technical capability, quality and after sales services.
There is a large literature associated with decision sciences, in which techniques for aiding or actually making decisions are proposed. Of most relevance to the present application is Multiple Criteria Decision Analysis (MCDA), which is an important area of decision sciences wherein scientific methods are investigated and developed in order to support decision making with multiple criteria.
A decision associated with multiple criteria is deemed to be properly made if all criteria in conflict are properly balanced and sufficiently satisfied. A MCDA problem can be generally modelled using a decision matrix, where a column represents a criterion, a row an alternative decision, and an element the outcome of a decision on a criterion. The decision matrix for a car selection problem, for example, may look like Table 1.
TABLE 1Example of Decision MatrixEngineGeneralSizeFuel ConsumptionPrice. . .QualityCar 11400 cc40 miles/gallon£8,000. . .GoodCar 21500 cc45 miles/gallon£9,000. . .Excellent..................Car N1100  47 miles/gallon£7,000. . .Good
Several methods have been proposed to deal with MCDA problems represented in the form of a decision matrix. Multiple criteria utility (value) function (MCUF) methods are among the simplest and most commonly used (see, for example, E. Jacquet-Lagreze and J. Siskox, “Assessing a set of additive utility functions for multicriteria decision making, the UTA Method”, European Journal of Operational Research, Vol. 10, pp. 151-164, 1982, and R. L. Keeney and H. Raiffa, Decision with Multiple Objectives: Preference and Value Tradeoffs, John Wiley and Sons, New York, 1976).
The MCUF methods are based on the estimation of utility for each outcome in a decision matrix. However, if a MCDA problem involves a large number of criteria and alternative decisions, estimating the utilities of all outcomes at every alternative on each criterion will become a tedious procedure and as such the MCUF methods will be difficult to apply in a satisfactory way (T. J. Stewart, “A critical survey on the status of multiple criteria decision making theory and practice”, OMEGA International Journal of Management Science, Vol. 20, No. 5-6, pp. 569-586, 1992).
Pairwise comparisons between pairs of criteria were primarily used to estimate relative weights of criteria in several methods including the eigenvector method (T. L. Saaty, The Analytic Hierarchy Process, University of Pittsburgh, 1988), the geometric least square method (G. Islei and A. G. Lockett, “Judgmental modelling based on geometric least squares”, European Journal of Operational Research, Vol. 36, No. 1, pp. 27-35, 1988) and the geometric mean method. Pairwise comparison matrices have also been used to assess alterative decisions with respect to a particular criterion such as in Analytical Hierarchy Process (AHP) (Saaty, ibid) and in judgmental modelling based on the geometric least square method (Islei and Locket, ibid). However, using pairwise comparisons to assess alternatives may lead to problems such as rank reversal as within the AHP framework (V Belton and T Gears “On a short-coming of Saaty's method of analytic hierarchy”, OMEGA, vol. 11, No. 3, pp 228-230, 1981; Stewart, ibid). These difficulties have lead to a long debate on how quantitative and qualitative assessments should be modelled and aggregated. Furthermore, both MCUF and AHP methods are incapable of properly coping with decision problems with missing information. If assessment information is missing for one criterion, one has to either abandon this criterion altogether or make assumptions, i.e., to use fabricated information. However, this may mislead the decision making process.
Fuzzy sets based methods have been developed to deal with MCDA problems with uncertainties. The main feature of such methods is their capability of handling subjective judgements in a natural manner. Therefore, they provide attractive frameworks to represent qualitative criteria and model human judgements (R R Yager “Decision-making under various types of uncertainties”, Journal of Intelligent and Fuzzy Systems, Vol. 3, No. 4, pp 317-323, 1995). However, fuzzy set methods suffer from two fundamental drawbacks. Firstly, they use a simplistic approach and limited linguistic variables to model a variety of information including both precise numbers and imprecise judgements. The consequences of this modelling strategy include the loss of precision in describing precise data and the lack of flexibility in capturing the diversity of information. The second drawback results from the use of fuzzy operations for criteria aggregation. Traditional fuzzy operators may lead to the loss of information in the process of aggregating a large number of criteria (J Wang, J B Yang and P Sen “Safety analysis and synthesis using fuzzy sets and evidential reasoning”, Reliability Engineering and Systems Safety, Vol. 47, No. 2, pp 103-118, 1995).
The present inventors have developed a MCDA method which has been termed evidential reasoning (ER) (see J. Wang, J. B. Yang and P. Sen, “Safety analysis and synthesis using fuzzy sets and evidential reasoning”, Reliability Engineering and System Safety, Vol. 47, No. 2, pp. 103-118, 1995, J. B. Yang and M. G. Singh, “An evidential reasoning approach for multiple attribute decision making with uncertainty”, IEEE Transactions on Systems, Man and Cybernetics, Vol. 24, No. 1, pp. 1-18, 1994; J. B. Yang and P. Sen, “A general multi-level evaluation process for hybrid MADM with uncertainty”, IEEE Transactions on Systems, Man, and Cybernetics, Vol. 24, No. 10, pp. 1458-1473, 1994; and Z. J. Zhang, J. B. Yang and D. L. Xu, “A hierarchical analysis model for multiobjective decision making”, in Analysis, Design and Evaluation of Man-Machine System 1989, Selected Papers from the 4th IFAC/IFIP/IFORS/IEA Conference, Xian, P. R. China, September 1989, Pergamon, Oxford, UK, pp. 13-18, 1990).
In the ER approach, it is proposed to use the concept of belief degrees in an assessment framework to model subjective judgements and develop an evidential reasoning algorithm to aggregate criteria in the assessment framework (Zhang, Yang and Xu; ibid: Yang and Singh, ibid; Yang and Sen, ibid). Compared with fuzzy sets methods, the ER approach provides a more flexible way of modelling human judgements (Yang and Sen, ibid) and the ER criteria aggregation process is also based on the rigorous Dempster-Shafer theory of evidence (G. A. Shafer, Mathematical Theory of Evidence, Princeton University Press, Princeton, USA, 1976, the contents of which, together with the contents of the other publications cited above, are hereby incorporated by reference). However, the prior art ER technique as described in the above mentioned publications is primarily of academic interest, since it is unable to properly accommodate a variety of “real life” situations. For example, the prior art technique is not capable of accommodating precise data or properly handling incomplete information, which may be caused due to a lack of information, the complexity of a decision problem and the inability of humans to provide precise judgements. Also, the old ER algorithm does not provide a rigorous process of aggregating incomplete information.
Therefore, there is a need to provide an improved MCDA technique which is capable of dealing with “real life” situations, and of overcoming the above described problems associated with the prior art.